Why is math research important?

As I’ve already described, I’m worried about the oncoming MOOC revolution and its effect on math research. To say it plainly, I think there will be major cuts in professional math jobs starting very soon, and I’ve even started to discourage young people from their plans to become math professors.

I’d like to start up a conversation – with the public, but starting in the mathematical community – about mathematics research funding and why it’s important.

I’d like to argue for math research as a public good which deserves to be publicly funded. But although I’m sure that we need to make that case, the more I think about it the less sure I am how to make that case. I’d like your help.

So remember, we’re making the case that continuing math research is a good idea for our society, and we should put up some money towards it, even though we have competing needs to fund other stuff too.

So it’s not enough to talk about how arithmetic helps people balance their checkbooks, say, since arithmetic is already widely known and not a topic of research.

And it’s also a different question from “Why should I study math?” which is a reasonable question from a student (with a very reasonable answer found for example here) but also not what I’m asking.

Just to be clear, let’s start our answers with “Continuing math research is important because…”.

Here’s what I got so far and also why I find the individual reasons less than compelling:

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1) Continuing math research is important because incredibly useful concepts like cryptography and calculus and image and signal processing have and continue to come from mathematics and are helping people solve real-world problems.

This “math as tool” is absolutely true and probably the easiest way to go about making the case for math research. It’s a long-term project, we don’t know exactly what will come out next, or when, but if we follow the trend of “useful tools,” we trust that math will continue to produce for society.

After all, there’s a reason so many students take calculus and linear algebra for their majors. We could probably even put a dollar value on the knowledge they gain in such a class, which is more than one could probably say about classes in many other fields.

Perhaps we should go further – mathematics is omnipresent in the exact science. And although much of that math is basic stuff that’s been known for decades or centuries, there are probably many examples of techniques being used that would benefit from recent updates.

The problem I have with this answer is that no mathematician ever goes into math research because someday it might be useful for the real world. At least no mathematician I know. And although that wasn’t a requirement for my answers, it still strikes me as odd.

In other words, it’s an answer that, although utterly true, and one we should definitely use to make our case, will actually leave the math research community itself cold.

So where does that leave us? At least for me straight to the next reason:

2) Continuing math research is important because it is beautiful. It is an art form, and more than that, an ancient and collaborative art form, performed by an entire community. Seen in this light it is one of the crowning achievements of our civilization.

This answer allows us to compare math research directly with some other fields like philosophy or even writing or music, and we can feel like artisans, or at least craftspeople, and we can in some sense expect to be supported for the very reason they are, that our existence informs us on the most basic questions surrounding what it means to be human.

The problem I have with this is that, although it’s very true, and it’s what attracted me to math in the first place, it feels too elitist, in the following sense. If we mathematicians are performing a kind of art, like an enormous musical piece, then arguably it’s a musical piece that only we can hear.

Because let’s face it, most mathematics research – and I mean current math research, not stuff the Greeks did – is totally inaccessible to the average person. And so it’s kind of a stretch to be asking the public for support on something that they can’t appreciate directly.

3) Continuing math research is important because it trains people to think abstractly and to have a skeptical mindset.

I’ve said it before, and I’ll say it again: one of the most amazing things about mathematicians versus anyone else is that mathematicians – and other kinds of scientists – are trained to admit they’re wrong. This is just so freaking rare in the real world.

And I don’t mean they change their arguments slightly to acknowledge inconvenient truths. I mean that mathematicians, properly trained, are psyched to hear a mistake pointed out in their argument because it signifies progress. There’s no shame in being wrong – it’s an inevitable part of the process of learning.

I really love this answer but I’ll admit that there may be other ways to achieve this kind of abstract and principled mindset without having a fleet of thousands of math researchers. It’s perhaps too indirect as an answer.

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So that’s what I’ve got. Please chime in if I’ve missed something, or if you have more to add to one of these.

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It is perhaps true that no mathematician does research for the sake of solving real-world problems (although I might be an exception), but that does not invalidate the argument, and I don’t know any mathematician who would object to real-world utility (aside from pacifists like Hardy, Grothendieck, etc.).

What is true is that, even if mathematicians themselves are not motivated by real-world applications, the applications inevitably benefit from math research just the same. Einstein would have had a much harder time doing general relativity if differential geometry didn’t exist, even though physics was never the main motivation for differential geometry.

“In 1912, Einstein returned to Switzerland to accept a professorship at his alma mater, the ETH. Once back in Zurich, he immediately visited his old ETH classmate Marcel Grossmann, now a professor of mathematics, who introduced him to Riemannian geometry and, more generally, to differential geometry. On the recommendation of Italian mathematician Tullio Levi-Civita, Einstein began exploring the usefulness of general covariance (essentially the use of tensors) for his gravitational theory. For a while Einstein thought that there were problems with the approach, but he later returned to it and, by late 1915, had published his general theory of relativity in the form in which it is used today.”

Differential geometry is the math you need to write equations saying how matter curves space and time. Trying to learn Einstein’s theory of this is what got me (and many people) started on learning Riemann’s wonderful work.

“it’s an answer that, although utterly true, and one we should definitely use to make our case, will actually leave the math research community itself cold.” It doesn’t leave ME cold! At any rate, I’m not sure it matters: “Why you want to do math” and “Why someone should pay you to do math” are two different questions, and I’m untroubled by their having two different answers. Lots of people build companies because they want to get rich, not because they want me to have a nice product; but the reason I give them money is because I want the product, not because I want them to get rich.

Gordon Henderson

February 13, 2014 at 8:55 am

Cathy,
Your points might all be valid, but I’m not sure how they lead to an argument for funding maths research:
– If maths research leads to solutions to real-world problems, then presumably private capital would be available to fund it, and public funding is therefore unnecessary.
– Maths does have an intrinsic beauty, but you’re right when you say it’s inaccessible to most. I don’t think it’s reasonable to ask “everyone” to fund something that only “some” will find beautiful…. and if “everyone” thought it was beautiful, presumably they would be willing to fund it privately?
– Maths might train people to have a skeptical mindset, but so do many other academic and professional disciplines. Off the top of my head, I can see elements of this in law, medicine, economics, philosophy…..and again, it’s not clear why society should fund training for individuals in pursuit of skepticism.
In fact, it’s hard to see why maths research is so different to any other kind of research; you might be better off embracing that, and making a more generalised argument in favour of funding academic research generally. Those arguments are a little more robust in terms of the “public/private” discussion, and aren’t damaged by a refutation of the case that math is more special than all the other disciplines crying out for funding.

The reason math is different (and the reason it deserves more attention w.r.t. funding) is that most people don’t have the faintest clue what it is that mathematicians do! It’s easy to fund research in computer science and medicine and economics because they give tangible descriptions of what their goals (write fast programs, cure cancer, fix the economy).

But pure mathematics research is very difficult to fund because its applications are almost never immediate, but when applications are found they’re almost always earth-shattering.

You can’t trust a corporation to have that kind of foresight. And it almost *never* happens that someone trained in a different science converts to mathematics, so you can’t trust other sciences to build the kind of mathematical thinking we need to make progress in the subject. Indeed, I’m reminded of a recent paper in biology where the authors unwittingly re-derived the trapezoid rule claiming it to be a new discovery. They’re about 200 years too late, and an introductory calculus class short!

“If maths research leads to solutions to real-world problems, then presumably private capital would be available to fund it, and public funding is therefore unnecessary.”

Oh! So we don’t need the government to build roads, enforce food and safety laws, or run a police force! (These solve real-world problems too.)

Gordon Henderson

February 18, 2014 at 6:09 pm

Funnily enough, roads are a great example. There are a lot of roads that funded by tolls – so that the people who think they’re useful pay for their upkeep. There are also plenty of roads internationally that are built using private capital.

DJ

February 18, 2014 at 6:19 pm

Your argument was that public funding for math/roads/whatever is unnecessary. It is not enough to argue that there exist roads built using private funding. Clearly, not every road is built using private funding. Do you think public funding for roads is totally unnecessary?

Gordon Henderson

February 18, 2014 at 7:52 pm

No, I don’t think that public funding for roads (or maths) is unnecessary; my point is that an argument for funding maths research on the basis of its economic utility is a weak one, since obviously valuable research would be funded privately. One can construct various arguments for funding maths research, but the strongest ones are likely to involve a generalised case for funding academic research generally rather than a narrow pleading that “maths is special, and more special than biology/physics/literature/whatever”.

Peter

March 16, 2014 at 2:39 pm

Just as there is much privately funded math research.

Thomas Ball

February 13, 2014 at 9:00 am

Is math all that different from other disciplines in its academic job potential? As a profession it seems healthy enough as the demand for the mathematically and technically trained continues to grow. Moreover, it would seem that very few departments, if any, can offer much certainty of employment in the university beyond the adjunct level.

Many highly technical professions face the dilemma that what they do is misunderstood by the typically innumerate layman, if its understood at all. In response, groups like the ASA have mounted public awareness campaigns in an effort at raising the level of “statistical literacy” in the general public. Personally, I think this is a misguided waste of resources. Professional associations might be better served to focus instead on raising their rank and file’s level and ability to communicate to the nontechnical.

Maybe this is already well-known to people here — but there’s philosophical research exploring the ways that bits of mathematics that were developed for aesthetic qualities and not for practical purposes then turned out to be immensely useful in applications. This is one theme of Mark Steiner’s work. This might at least lessen the oddness of making the case for math based on its real-world applications, even as we encourage mathematicians to pursue what they find most beautiful, intriguing, etc.

Also, I think many great mathematicians of old (like Newton, Euler, Gauss, and Lagrange) created math for the explicit reason to solve real-world problems. Maybe that impulse is less common these days, but some are still motivated in this way, e.g., Dirac, Shannon, or more contemporaneously, Daubechies.

Deane

February 13, 2014 at 9:21 am

I think you’re raising an extremely important issue, especially for the math community, since it is a crucial factor in building support for math, financial or otherwise, in the community at large.

Reasons #1 and #2 are the most common ones I see, and I don’t find either particularly convincing. I also sense that most people don’t and react skeptically when we say them. Reason #2 is also particularly unattractive to me, because it puts us with liberal and fine arts, which are viewed as being non-essential and therefore are poorly funded.

I find reason #3 to be the most convincing. To me it leads to a more concrete explanation of why mathematics matters and distinguishes us from other fields. I consider mathematics to be the development of new knowledge (from what we already know) using abstraction and deductive logic. Although other fields certainly use these tools too, mathematicians take it much further and can do it in a way that cuts across different fields.

One example I like to use is diffusion and stochastic processes and specifically the heat equation. I don’t know where this equation was first used, but eventually it appeared in several different contexts, including physics, chemistry, and finance. Mathematicians were able to see that a single abstract idea applied to these different situations, develop their understanding of it, and transfer their knowledge back to the specific applications.

And sometimes the abstraction comes first. Sometimes mathematicians develop the abstractions themselves, and then the applications come later. The two obvious examples are differential geometry applied to special and general relativity and number theory applied to cryptography.

I’d also like to say a word about using math as a “BS detector”. Unfortunately, in my experience this is a rare skill that is taught better in other fields such as physics and engineering, where “back-of-the-envelope” reasoning and calculations are taught more. A simple example is “unit analysis”, which in mathematics would be called “scale invariance”. Although I see some mathematicians quietly using such techniques themselves, I don’t think I’ve ever seen it mentioned explicitly in a textbook or taught in a lecture.

Too many mathematicians, especially students, rely too heavily on sophisticated theorems and machinery to answer questions and problems. They don’t start with simple examples and approaches first and bring in heavier machinery only as needed.

This has led me to conclude that we don’t teach math well at any level. A lot of attention is focused on elementary to undergraduate math education, but I haven’t seen the math community ask itself whether it is training Ph.D. students well or not. When I interviewed math Ph.D.’s for jobs in finance and calculus, I was shocked by their inability to solve simple problems, even though they could rattle off very fancy theorems and their proofs.

I think that the best way to make a case is to give one example. For instance, you can take the example of fractals:
0) Fractals are highly counter-intuitive objects which could not have been discovered without research for aesthetic mathematics. For centuries, they were out there but no one couldn’t formalize them. Only paid mathematicians could have discovered it, as it takes time and builds upon complex foundations of mathematics (although, granted, Mandelbrot was working for IBM…).
1) Applications are countless. My favorite one regards phone receptors. If it weren’t for fractals, telecommunication would still be made with wires or big paraboloids.
2) Do I really have to insist on that? You may simply take the example of Pixar, which exploits fractals to make stunning animation movies.
3) Mandelbrot was rejected by the math community at first. It shows how hard it is to embark on his ideas. Yet, once we do, the world suddenly looks totally different. We see more patterns emerging. Learning fractals is about learning how to look at the world differently, in both a more rigorous and more daring way. And I’d argue that only mathematics can teach that so radically.

jms

February 13, 2014 at 9:47 am

To answer your question I think you have to take a broader point of view and ask (1) why is research in any field of fundamental knowledge (vs applied science) necessary, (2) what do student get from a university education?
For (1) you can think of theoretical physics, philosophy, humanities, etc.
I think the answer to (1) is that interacting with people who are expert at one of those abstract areas of knowledge is a great formative experience for students. It’s completely transverse to learning eg how to solve an ODE or integrate a function (which can be useful, too).
Of course you could train student in calculus without ever meeting anybody who has done any research in mathematics. But experience shows that this would quickly turn into a set of boring exercises with little intellectual value. (Ok a calculus class can be like that, but not always, and higher-level math courses tend to be much more interesting.)
The same could be said of the other fields mentioned above, but perhaps mathematics has a special role since it helps acquire some qualities (abstract thinking, rigor) not easily found together in other fields and important in cultural and intellectual life, and sometimes even in daily life.

Edward Frenkel recently gave a talk at MOMA PS1 entitled “Art, Love and Math.” He believes in a symbiotic relationship between artists and mathematicians. From the mathematician’s point of view this is nothing new. Poincare’s quote comes to mind: “A scientist worthy of his name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.” It does sound a bit elitist, but I am sure that anyone who studied math agrees, at least in part. What Frenkel is trying to do is to bring some symmetry into this equation by saying that artists can help mathematicians inform their work, to see things in a new way. I am neither an artist nor a mathematician to have a strong opinion here, but I think all of us can agree: symmetry is a beautiful thing.

Ben Braun (@BraunMath)

February 13, 2014 at 10:16 am

The National Research Council report “The Mathematical Sciences in 2025″ is a very helpful resource for gathering information, data, arguments, etc in support of mathematics research. It is free to download here: http://www.nap.edu/catalog.php?record_id=15269

A minor thing to maybe add onto point 2: math discoveries seem (to us humans) timeless and can be further developed (for fun, not necessarily for practical applications) by future peoples. Work that is the result of funding now could continue to be used and improved by generations on end. Also if our civilization dies out, but another human one arises, chances are the math we discover will now could continue to be advanced by them. So I guess if we wish to leave a legacy, or some parting gift, besides chocolate, to future peoples, we should fund math research. one reason for throwing in the word human is that math, like art, is a by product of the workings of the human brain. Superior intelligent beings may not be so amused by our maths. I encountered the following quote after your post yesterday led me to the blog mathtango:

“I have come to believe, though very reluctantly, that it [mathematics] consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-legged animal is an animal.”
— Bertrand Russell (1957)

Or turn the question on its side by admitting that a lot of math research being done may not be important in terms of applications, and is an activity of leisure. A sign of a healthy(?) or at least affluent society, is willingness to spend on things that are not 100% necessary…ample discretionary spending. I don’t know if our society will stay (or is) affluent given all sorts of gloom and doom we face, but hey we should at least try keeping up appearances. Also somewhat in the same vein, spending is good for the economy…it keeps math professors employed, who are people too. But I guess you are saying there aren’t going to be math professors to begin with.

a comment on point 2: the public (and even fellow mathematicians) can’t appreciate contemporary mathematical advances directly, but mathematicians can help them appreciate it through popularization efforts… (I forget the french word but isn’t it something like vulgarisationes?) It’s fun/inspiring/worth paying money to see the feats the human brain can achieve. But your point is still valid…we would just be hearing stories of feats, like stories of LeBron James dunking or whatever, rather than being witnesses. And we would have to accept those feats are mathematically correct, based on the testimony of a few experts.

Zathras

February 13, 2014 at 1:44 pm

I don’t think it is useful to talk in absolutes here. Very few people will dispute that math research has great value. The issue here is the distribution of resources. So the question is not just whether math research is good, but whether math research is better than research in other fields. This is a more uncomfortable argument to make, but a necessary one. The link of teaching supporting research has benefited math tremendously, even though there is no real correlation between teaching value and research value. Why should math get money and others not? In a budget-constrained world, that’s the argument that has to be made.

DJ

February 13, 2014 at 2:55 pm

I think many people, especially those not active in research, misunderstand the nature of the link between teaching and research. Contrary to what you might expect, it has nothing to do with any sort of imagined correlation between teaching value and research value. The best researchers are often horrible teachers and vice-versa. Instead, the crucial synergy is that active teaching, at any level of competence, helps advance your research, and vice-versa. We ask top researchers to teach, not because they are particularly good teachers, but because the act of teaching is something that helps to benefit and advance their research agenda. Teaching forces you to crystalize and augment your insight into the fundamental issues of your field. It also offers a ready-made source of slave labor, er I mean graduate students. Similarly, active research, at any level of ability, gives you an unparalleled perspective into the evolution of your field, a perspective which is invaluable to have, even when you’re just teaching first-year calculus. No, the best teachers are not the best researchers, but if we forcibly separated the two activities, I guarantee you, the best teachers would get worse, and the best researchers would get worse. I myself worked in a research-only position for many years, and I had the opportunity to stay in such a position, but I chose the life of a professor because I came to realize the importance of linking teaching and research in my life. Nowadays I am a mediocre researcher, but an outstanding teacher. My middling research agenda is nevertheless something that my students find extremely fascinating, and for many of them it is a big factor in keeping them interested in my field.

Concerning your other point, about distribution of resources, I don’t think research investment is a zero-sum game. Research pays for itself by creating new opportunities. If you wish to compare costs, that’s fine, but please be clear that the correct comparison must be made using opportunity costs, not budgetary costs.

Zathras

February 13, 2014 at 3:22 pm

“Research pays for itself by creating new opportunities.”

I don’t disagree, personally. However, the people who have to make an annual budget have to make the distribution. That’s just the way the world works. And that means comparisons have to be made.

Reblogged this on CancerEvo and commented:
Is math education important? Why is that even a question? Nice post.

Howard Eaton

February 13, 2014 at 3:42 pm

Math research that is aimed directly at some particular use or problem that arises in a physical or social science, or in business, industry, or government, is likely to be funded by someone. I think the real concern is about math research that doesn’t have any obvious use, and therefore is likely undertaken by the researcher out of his own interest.

Two points:

1. Whether mathematicians like it or not, this kind of mathematics research is a humanity, not a science. Like music, painting, sculpture, literature, philosophy, people do it because of its intrinsic fascination, for them, out of an urge to solve interesting problem, discover and explore new territories. So whatever funding society chooses to allot to these as culturally significant activities, this kind of math research will have to be content with its proportionate share.

2. The concern about elites is misplaced, if we understand what is meant. True, very few will understand the significance of most of this kind of research. I had a career teaching literature. What proportion of the population reads the great works of literature, once they leave the university? Some, but in North America for sure, not a high proportion. Great works of literature often require a lot of effort, and sometimes some preparation. They are often intellectually, emotionally, morally challenging. They aren’t particularly restful, easy diversions. Those who are occupied with work, family, community affairs, those who are tired at the end of a long day, will probably not find much more or less mindless comfort in WB Yeats or William Faulkner. There is no inferiority here, just the pressure of ordinary people with ordinary lives, whose elite interest may be automobiles or football statistics.

Are those who do invest time and energy in great books, an elite? Yes, but so are those who appreciate opera (I’m not one), classical music (I am one), architecture, garden design, fine furniture, or fashion design. So far as I can see, these are more than merely harmless elites, they are carriers of culture, of refinements of knowledge. Membership is open to anyone for the most part, and membership does not confer any significant amount of unearned and dangerous power or privilege. Such elites are good, not bad.

There is contamination of the term elite from its association with other social elites–political, financial, corporate, etc,, which are deservedly suspect, and always dangerous if not always guilty of anything.

A final remark about the way many math researchers are supported in university settings. Most math instruction in calculus and statistics, in the first two years of university, was carried out in my day, by math graduate students. This was not a success. The highly intelligent persons who are drawn to mathematics research of the kind we are discussing, are more often than not, not especially gifted in human relations, not especially articulate, not very well equipped to understand why what is so blindingly obvious to anybody, presents such problems for undergraduate students. Forcing math researchers to do a lot of basic undergraduate math instruction, in order to earn the time they need to do their research, seems to me a dubious bargain.

I am a great admirer of Richard Feynman. He was a great lecturer in my view, very inspiring to me personally. But many of those who took his courses found them to be very little useful in their practical problem-solving efforts in engineering, and even in physics. Profoundly devoted to physics, he took charge of his own learning from an early age, and never had much understanding of or patience with, plug-and-grind students who wanted a career doing useful and profitable things, not a deeper understanding of the physical world. It seems to me that the great researchers should be supported in their work, to the degree we can afford it, but we shouldn’t impose them on the bulk of undergraduates, unless they happen also to be interested and well equipped to do this kind of work.

DJ

February 13, 2014 at 6:35 pm

1. I am not concerned about math that has obvious applications, and I am not concerned about math that has no use. These extremes will, as you say, take care of themselves. My greatest concern is for those areas of math research in the middle, which have no obvious immediate use, but which later (often much later) turn out to have critically important applications. The poster child for this phenomenon is my own research area: number theory. For hundreds of years, this branch of mathematics was widely regarded as the purest of the pure, the most abstract of all mathematical thought. G.H. Hardy went so far as to say that number theory would never be used as an instrument of war. And then, the internet happened, and people suddenly needed public-key cryptography, which relies critically on number theory, and the NSA got involved, thus defying Hardy’s prediction. Nowadays it is no exaggeration to say that the entirety of internet commerce depends on number-theoretic primitives discovered 200 years ago, at a time when number theory was unquestionably a pure humanity subject.

It is not easy to predict which areas of mathematics will turn out to be important in the future. If we try to fund math at the level of humanities subjects, untold future opportunities will be lost.

2. I think you vastly underestimate how inaccessible math research is, compared to other arts. A single masterpiece like Wiles’s proof of Fermat’s Last Theorem or Mochizuki’s proof of ABC can easily take 5-10 years to fully understand. Probably no more than 1000 people in the world can claim full understanding of Wiles’s proof. Probably no more than 1 person can claim full understanding of Mochizuki’s proof (and yes, the lack of people is obviously an issue). I don’t know of any other academic subject, really, where average journal papers routinely take a year to read. Also, other arts have ways of being made more accessible; for example, I find it excruciatingly challenging to read Shakespeare’s plays, but in a live performance they are pure joy. It’s really hard to make math accessible.

3. As I explained in an earlier comment, teaching in universities makes much more sense once you realize that teaching duties are largely for the benefit of the instructor, not the students. Since you brought up Feynman, I might as well use him as an example. To quote from Feynman’s own words:

‘I don’t believe I can really do without teaching. The reason is, I have to have something so that when I don’t have any ideas and I’m not getting anywhere I can say to myself, “At least I’m living; at least I’m doing something; I’m making some contribution”–it’s just psychological. So I find that teaching and the students keep life going, and I would never accept any position in which somebody has invented a happy situation for me where I don’t have to teach. Never.’

It should be clear from the above that the main purpose of Feynman’s teaching duties was not to benefit his students, although some students may well have benefited. He would have taught classes even if it came at a cost to his students, since he more than made up for it with the increased research productivity that resulted from having nonzero teaching duties.

Howard Eaton

February 14, 2014 at 3:56 pm

A. “I am not concerned about math that has no use. . . . Nowadays it is no exaggeration to say that the entirety of internet commerce depends on number-theoretic primitives discovered 200 years ago, at a time when number theory was unquestionably a pure humanity subject. It is not easy to predict which areas of mathematics will turn out to be important in the future. If we try to fund math at the level of humanities subjects, untold future opportunities will be lost.”
It seems to me that from your own example, you are exactly concerned with math research that has no apparent use. I’m not a number theory person, but I think these primitives are what I learned to call primes. Mathematicians 200 years ago, who found these interesting and did a lot of work with them, had no more idea they would be “useful” than the man in the moon. No one in his right mind is going to fund math research on the off chance that some result will find a use we can not even imagine, 200 years, or 2000 years, in the future. And the truth is, most of it will find no use, and will be of interest mainly, or only, to mathematicians. It has net present value of 0, as a practical activity, to a good first approximation.
When you say, “fund math,” what do you mean? I would be willing to bet that the great majority of math funding is for instruction or research in “useful” mathematics, perhaps calculus, statistics, linear algebra, for undergradutes, and graduates in science and business, instruction and research that has important present applications. This is bound to continue. And I think we agree that the funding for research in applied math, will continue, and perhaps even grow. None of this is funding for the kind of research you are discussing.
B. A large proportion of this kind of pure research in a university setting, is supported by money which legislatures and students paying tuition, imagine is going to support instruction in the “practical” mathematics (discussed above), they can see the value in. This brings me to a point where we may well be distinctly on opposite sides.
Let’s consider your comment: “[T]eaching in universities makes much more sense once you realize that teaching duties are largely for the benefit of the instructor, not the students,” and the Feynman example.
If I can rephrase a bit, I think this means that teaching duties are a purely secondary duty, that the faculty member and the institution are far more concerned about research and reputation, than responsible, competent, successful work on the teaching task. I believe you are right, that this is in fact the case. I think this explains why math and science are among the worst taught subjects, why they are becoming increasingly unpopular choices, where students have a choice.
I also think it is a profoundly dishonest and corrupt practice. If students (and legislatures and the general public) should decide to burn the university down, that is prepared to sacrifice the education of thousands and tens of thousands, for the benefit of a few scholars and their departments, I will happily supply the matches. The worst consequence of this kind of corruption is that many otherwise able students are driven out of math and science by ineffective, incompetent instruction, long before they have any chance to discover how interesting the subjects are. And this especially applies to women.
To reduce teaching to a recuperative, inconsequential pastime for scholars passing through an unproductive period, and the education of students to a sacrificial pawn in the pursuit scholarly ambition, is socially irresponsible and morally indefensible.
When I look at Feynman’s lectures, both his university lectures and his more public ones–say the lectures he gave at Aukland on quantum electrodynamics–I don’t think we are dealing with your ordinary physics course. I would argue that, at his best, Feynman is
making physics interesting, and accessible. He communicates his sense of the excitement in the project of understanding the physical world. For that reason, the lectures are still good decades later. And for that same reason, they weren’t all that useful to those who wanted to get on with the mastery of basic skills. In my opinion, the value of lecturing for Feynman, came from trying to bring clarity and coherence into his own thinking, putting everything into a broader perspective, which is no doubt useful for anyone who works a lot on very specific problems. Beyond that, Feynman was a showman, he enjoyed an audience, and he loved to talk about physics. By all means, put him in a lecture theater and let those students have a chance to be excited, intrigued, inspired by the achievements and challenges of physics. But make sure that someone with time, patience, an interest in people, a desire to understand how students best learn the basics of physics, and a primary commitment that they do learn them as easily and efficiently as possible, takes care of the rest. This is not a job for the Feynmans of this world. I would guess that learning any subject is 10% inspiration, and 90% perspiration. I doubt that the person who provides the first, is very often the best person to manage the second. There is a lot more work for the second type, than the first.
It is not very difficult to imagine that there might be many activities other than teaching undergraduates, which could occupy a serious researcher who is having a dry spell. Feynman painted, played drums, and sometimes picked a whole new line of research and exploration in physics, for something to do. And why not? If researchers want to teach, if they have the talent and skills for it, great. To force the others, perhaps the majority, into doing so, seems perverse, at the very best. Let them find other activities that don’t require a major betrayal of trust. (Note: I exclude graduate teaching within a departent from my consideration. There are some legitimate concerns about the line where apprenticeship ceases, and slave labor begins. Doesn’t concern me here. For one thing, the volume of graduate teaching within a department, is fairly small, in relative terms.)
C. Your point 2, accessibility. I have to agree, that appreciating a lot of math research is very hard. Some, perhaps much, may never be very accessible, except to other mathematicians. And the number of people interested will always be a small proportion of the general population. But what is maybe more important is that there is a somewhat larger elite with a general sense that mathematics is worthwhile, if a fair chance at funding is what is wanted. Here I think biographies of important mathematicians, explaining in a necessarily simplified way, their contributions, would be good. And there are several good histories of mathematical thinking and discoveries. And frankly, a bigger effort by mathematicians, and maybe some mathematical journalists, to explain some of the contemporary directions in mathematical research, might help. I’m sure your blog helps.
Also, I don’t know if you appreciate how difficult some modern and contemporary art is, how objectionable it can be to some people, and how much resentment there can be, when public money is spent on it. Your Shakespeare example is not the best. You need to remember that most of Shakespeare is drama, plays: it was meant to be seen as a play. It’s true that university courses tend to deal with the text, because they find it easier to present. It is easy to get a cheap text and read the play. This economy comes at a high price. Drama, and much of poetry, is performance art, not a mass of word games and literary devices, with the meaning to be puzzled out from an obscure text. If you were interested in classical music, do you think it would make lot of sense to look mainly or only at musical scores, and not hear it played by a skilled orchestra?
Better examples would be, at one end, Chaucer, and at the modern end, Finnegans Wake. For Chaucer, you have to do a lot of language work, and some culture and history, before the bulk of the meaning is accessible. Not many will do it. Finnegans Wake is a reading experience so different from what people expect from a novel, that very, very few will ever read it. TS Eliot, and perhaps even more, Wallace Stevens, are classics in poetry in English, but considered very “difficult” by most.
Modern, abstract painting is so little understood, that there is often an uproar when a museum pays a lot of money for “what my ten year old could do.” Mathematics may be the densest and most difficult of all languages, and not easily appreciated, but I doubt it makes many positive enemies.
Feynman remarks somewhere that, if you meet someone who tells you he understands quantum mechanics, he doesn’t. In his Aukland lecture, he tells the audience that if they really want to understand what he is describing to them as clearly as they can, they can only do so in the language of mathematics. I am sure that is true. Bessel functions or else. I spent several months once working through the derivation of the key mathematical equations of quantum mechanics. The number of equations is small. The little book I used was only 80 pages long, but some pages took several days. So I will gladly agree that mathematical language is highly compressed, and highly refined, and makes a hard read. At the end, I understood most of the equations. But I don’t feel I understood quantum mechanics, meaning the underlying physical reality, any better. I doubt it can be understood, beyond the equations we have developed to describe it. I expect that metaphors and analogies from our macro experience do not make any sense in this micro world.
Having said this, Feynman was able to give a useful, accessible view into quantum electrodynamics. I am going to take a leap of faith, and suggest that the broad outlines of mathematical research in algebra, number theory, topology, etc can probably be made somewhat more accessible to that small part of the public that will be interested. Not every research project, but maybe some of the major discoveries and problems.

DJ

February 16, 2014 at 9:07 pm

I can’t respond comprehensively to every point you raise (it’s way too long), but I will make a point of correcting your misinterpretation of the relationship between research and teaching. The intent is not to corrupt education. Far from it. The main point is that the very best researchers need to teach more than zero students in order to maximize their research output. Similarly, the very best teachers need to conduct more than zero research in order to maximize their teaching success. It has nothing to do with preferring either one over the other, and your insinuation that this is the case is a massive perversion of the truth. Simply put, in a world dominated by specialization, it is worth investing the effort to make sure that your top performers perform at their best in their specialty, in any area, whether related to mathematics or not.

SM

February 17, 2014 at 9:01 pm

So… how much more than zero teaching do you find is useful for research? A standard 2-1 or 2-2 load? 4-4? One course a year?

DJ

February 18, 2014 at 6:57 am

I don’t claim to be a top researcher, but my feeling is that:

1-0 is enough for the purpose of psychologically justifying your paycheck during the periods when your research is not going well. This is probably the minimum requirement for most people.

1-1 or 2-1 is probably optimal for most people’s overall productivity, since it provides ample opportunities to recruit students into your research, without requiring too much time to be spent teaching.

Anything 2-2 or over means your teaching time will start cutting into your research output.

I think another possible argument (closely tied with the first, but possibly more in touch with mathematicians) can be extracted from an analogy I came up with not long ago. As anyone doing math, when talking to non-mathematicians about my work, the question “What is this useful for?” invariably comes up. It’s always struck me as a misguided question, but it took me a long time to come up with a reason. Eventually, I found this analogy:

We humans live in a “city” that is well-connected, with roads and buildings and houses. This city is “common knowledge.” Outside of the city is more specialized knowledge and even “possible things that are unknown”. On the outskirts of the city there are factories, responsible for the goods those of us in the city take for granted every day. Farther out, there are mines and logging camps and farms, and what-ever else. These “raw resources” are cutting-edge “industrial knowledge”–new communications technology, new medical technology, new energy technology, etc. Farther out still, there are areas with untapped resources, and areas that are even unexplored.

Engineers are the “miners” (extracting “raw knowledge” into something useful), scientists are “prospectors” (looking for “raw knowledge”), and mathematicians are “cartographers” and “explorers” (looking for where there is “potential knowledge”); The city is mapped out very well, and really, you can get around without a map–you mostly stick to neighborhoods you know, and when you’re lost, you can ask a local. The outskirts are also mapped out, but less well-known and less trafficked; having a map is often a good idea. Of course, we already have those maps.

But the wilderness, far outside the city… we often have maps to established mining camps, and usually the prospectors are following maps, but if we want to establish new mines (develop new technologies), and prospect new areas (understand our universe better), we can either stumble around in the wilderness, looking everywhere for things that may be of use, and hoping to find something of value. Or, we can have somebody go out and make maps—show us the easiest routes, describe the landscape, tell us where to build bridges or tunnels, and find rock formations that are likely to contain something worthwhile; we can have someone explore the world and make maps.

Without mathematicians, scientists are stumbling around blind in the wilderness, hoping to find something of value. Without scientists finding things of value, engineers are wasting their resources trying to extract value which may or may not be there.

[Generated with seriousness+snark but not REALLY snark:] National security, GNP, and inspiring the young, all without capital-intensive overhead. A strong nation is a mathematical nation, and there is no other area of basic research where dollars can go so far since test tubes, monkees and particle accelerators are not needed. The infrastructure that springs up around funding and the broad social regard for areas deemed important enough to fund significantly will have social networking effects, improving the status of mathematical research so it seems like a no-brainer for continued support. One never knows what breakthroughs will yield the most spectacular results, but these are incidental byproducts of nationally substantial basic research programs. Mathematics research has the incidental benefit that the personnel are most capable of being redeployed into other get-a-job areas when necessary. Additionally, the developing world is well aware of these advantages and in many cases have programs in place that outstaff American efforts. We must not fall behind.

Cathy, I think one good place to start investigating how to make your argument is to look at what makes mathematical research and mathematical thinking skills different from other scientific pursuits. This would help you focus on making your argument specific to mathematics.

For example: as you know mathematicians focus with immense scrutiny on finding the right definitions for a concept and making them rigorous enough not to be open to interpretation. This seems to me to be a unique feature of mathematics.

This means that mathematicians think about problems like no other scientist, engineer, or politician does. As a result, they can much more easily identify straw men and pointless endeavors, and better evaluate the potential for an idea to yield fruit.

Mathematics also enjoys the full benefit of all analytic tools because mathematics invented them! For example, Terry Tao recently wrote a blog post about using ideal fluids to construct logic gates and implement a computer. Ideas borrowed from drastically different places are common in mathematics, and part of why it’s so important to have people who are fluent in it.

Reblogged this on actuarialjourney and commented:
I’ve been trying to post a comment on this article from MathBabe

FH

February 13, 2014 at 7:28 pm

I don’t think 1 and 2 are separate arguments. Math research is useful precisely because it is driven by instincts other than merely application:

Mathematicians are explorers. Mathematical research is the act of exploring the vast space of the undeniably true. They explore and bring back proofs. That is, patterns of pure thought that are established beyond all doubt.

They don’t go out to find useful patterns. They go out to find interesting patterns. The patterns themselves drive them, their curiosity, the need to create. However the more patterns we know of, the more we can find and use in reality. These patterns, brought back with no application in mind have turned out to be incredibly powerful for understanding what is going on all around us. We often find mere threads, a few bits and pieces of color, and take them to the explorers: “Yes yes! I have seen this before. On a journey not so long ago I came across just such a pattern. Here let me fill in the pieces you were missing for you!”

We don’t know what’s out there. We don’t know what novel ways of thinking we might yet discover, or when someone will recognize these patterns somewhere out there. If we stop giving these explorers time to dream up new ways to think, we will overlook the patterns when we encounter them in reality. If we, as a species, give up on this purest form of a will to create, a will to know, to explore, without bounds the outer limits of what is humanly thinkable, we will be the poorer for it.

Is the notion that mathematics is the language of the universe too romantic? I think this point is separate from it’s aesthetic appeal, and instead makes a claim about the boundaries of human knowledge.

“Continuing mathematics research is important because it has always been an important tool for understanding the universe.” (And there’s still a lot of stuff that we don’t understand.)

The main assumption is that understanding the universe improves our quality of life, and that mathematics helps us do that. The same can be said of all of the sciences; they represent the areas in which we are pushing the boundaries of our understanding bit by bit. From there I think a strong argument can be made that pushing the boundaries of what we know has done more to improve human quality of life throughout modern history than any other activity. Furthermore, history has shown that we are really bad at guessing what research will end up being useful, and when we might realize it is useful, so we just have to do a lot of it.

Framing the math research issue as “is it important” (of course it is!) isn’t going to be very helpful. Influencing the funding debate requires addressing at least two issues:

How much (math) research is important?
How should it be funded?

By any reasonable metric, we’ve massively increased funding for many areas of academic research over the past few decades, including mathematics. Teaching loads have been reduced across the board. Promotion and tenure research expectations have increased. Most sub-flagship state institutions, formerly teaching schools, now have research missions. Same with the regional private schools. The number of research outlets and the number of refereed papers published have skyrocketed over this period.

Unfortunately, many of these new outlets have minimal standards and add little, if any, value to society. In the sciences, John Bohannon’s study that found over half the newer fee-based open access journals accepted fake papers. http://www.thecrimson.com/article/2013/10/16/study-science-journals-fake-research/ These types of outlets are frequented by professors at institutions with newly developed research missions, often desperate to meet publishing requirements that tend to focus on quantity over quality.

Would the newer mathematics publications have fared better? I really don’t know but I can say that the rise of mediocre outlets extends well beyond the sciences.

So, can we argue that the marginal real increase in funding over the past generation has been productive? Perhaps, but the arguments in this post and the comments above don’t address this. Ultimately, these “marginal” institutions will be first in line when research cuts come. Are their new research missions worth saving?

Has the marginal increase in (math) research over the past 20-30 years been worth the cost? What has funded this increase in research? Student debt. While traditional research institutions have student loan default rates well under 2%, universities that have recently increased research emphasis typically have higher default rates, sometimes approaching and exceeding 10%. A far higher percentage undoubtedly lives in distress caused by student debt.

Should schools with higher loan default rates that typically admit less well-qualified applicants have research missions? Are those missions worth driving a significant portion of their students into long-term debt servitude? This is not an easy argument to make. Absent new research funding sources, reducing our student’s debt burdens by increasing emphasis on teaching and away from research at sub-flagship state institutions makes a lot of sense.

MOOCs or no MOOCs there will be always be constraints on levels of research funding. Would it make sense to focus the vast majority of that funding on 100 or so designated “research” institutions as we did a generation or so ago? Alternatively, should we spread that funding to several hundred additional schools as we’ve tried to do in recent years?

Jim Birch

February 13, 2014 at 11:22 pm

Continuing math research is important because we don’t have a comprehensive mathematics of space-time yet. It probably needs a new branch mathematics and we may have to develop several to find one that fits.

I think that there needs to be more mathematical involvement in fields such as computer science, medicine, neuroscience, and so on… More so than on the mathematical field itself. The reason I say this is because abstract math has a tendency to be ahead of real world applications and when it comes down to it, the act of funding research in mathematics is in part reliant on its real world applications. Mathematical researchers need to be outsourced to other fields!

Case in point. We are starting to see the rise of intelligent machines in the form of Siri and Wolfram alpha. Of course these machines are squarely in the domain of computer science but the logic by which they opperate are mathematical. Having a mathematician working away at developing efficient logic systems better than the ones utilized today is where the money is heading methinks.

Lets take genetics. At the university of Utah there is heavy research going into developing mathematical foundations which describe biological systems http://www.math.utah.edu/research/mathbio/. This applies equally to neuroscience and emerging fields such as nanotechnology. The point here is that mathematical research by itself is like a city with shops and stores and houses but without the people. If the people are other fields, even as odd as graphic design, then funding mathematical research without an emphasis on these domain is a waste of money. That doesn’t mean that math for its own sake ought to be done away with but I do believe money would be better spent getting more mathematical research to link up, applied or not, with other fields.

Perhaps the biggest reason for more research in math is the maintainance and evolution of the civilization it’s created. Look at any device and we can trace the origins of its existence to some mathematical theory or theory derived from heavy use of mathematics. Because of this society has become a complex automated system running on limited resources. If we stop funding mathematical research we run the risk of stagnating this evolution before it can discover better resources to run our system on. This would spell disaster. As another example look at the recent financial crisis that’s still reverberating around the globe. It was due to humans taking advantage of a mathematical system, the economy, with disastrous consequences. Now researchers are developing methods to analyze the system and detect when a bubble is forming in the economy and pop it before it goes full dragon lord. To do this you need mathematical techniques.

In essence what I’m saying is there needs to be more research in mathematics which communicates with other fields. Even if its to communicate the pure things which are beyond applied application at the moment.

DS

February 14, 2014 at 12:54 pm

(First time commenting. Love your blog! (though I disagree with a few of the views and therefore learn much))

Perhaps it is true that considering the value of math research primarily in terms of its aesthetic value is elitist; sure, not too many people are in a position to appreciate it. But, why is this a bad thing or an unsatisfactory justification? It is not that the in-group denies people the means to appreciate the works they produce, on the contrary, nothing would make us happier. There is a clear, identifiable path to become someone who can appreciate the beauty of modern math, definitely clearer than the path to become someone who can appreciate say John Ashbery or Stockhausen. In fact, I’m pretty certain there are more people in the world who’ve enjoyed at least one work of Serre than there are those who’ve enjoyed than a major modern poet, like say Geoffrey Hill.

If math is art, it is found art. I happen to believe that some of the best works of found art consist of mathematical knowledge. I can’t think of math as artistic creation. While generally agnostic when it comes to theology, I can’t even think of math as created by God. If there’s anything that’s utterly uncreated, it’s math. So it’s art and not-art.

I only got a BS degree in math. I didn’t have the guts to even try for graduate school, and of course marketability of math skills begins at PhD level, so I had to settle for a high-school level “career” as just a temp. But I don’t entirely regret majoring in math. Mathematical knowledge is the only kind of knowledge I know of that can’t be faked. I remember about 2 years ago reading about some people who got a word salad published, crafted of the usual words like “postmodern” and “deconstruction” and the like, and nobody realized it was a hoax. Somehow I can’t imagine someone pulling off something like that on a math journal. Mathematical knowledge can’t be obsoleted, either. The theorems I managed to learn 30 years ago are still demonstrably theorems. I could spend real $ on some proprietary trade school named after a corporation that used to be referred to as a sovereign state, and study a proprietary discipline called something like “MCSE,” and for what? A better shot at a job? A job where? On the waterfront? Spoiler alert…

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In the movie On the Waterfront, the guy who does the hiring has a strict policy of hiring only those applicants who have borrowed money. I’ve always wondered if the real reason employers feel they have a legitimate interest in applicant credit history is a preference not for the most solvent, but for, if not the least solvent, those with the most debt to service. Likewise I’ve sometimes wondered whether at least some of the employers who drug test applicants have a policy of hiring only the ones who test positive. I’m thinking Enter the Dragon, in which an opiate dependent staff seems to be part of the business model. Obviously, real life isn’t anything like the movies. I shouldn’t get worked up about imaginary things.

Another paranoid delusion I have from time to time is wondering if the real reason student loan debt is not dischargeable in bankruptcy is because you can’t repossess an education. But as I said earlier, there’s a difference between the eternal truths of mathematics (and the generally generalizable knowledge in the liberal arts in general) and the created, and expire-able “bodies of knowledge” in disciplines such as “MCSE.”

hilbertthm90

February 16, 2014 at 1:03 pm

I really hate to pull this card, but I think I have to. I think your misunderstanding about what mathematicians do stems from “only” having an undergraduate degree in the subject. Once you start trying to do math research, you come to a different conclusion. Doing real math is an incredibly creative endeavor. At the undergraduate level it is still mostly about memorizing facts and theorems. Real math research is about producing new theorems. It is too long to post here what goes into this. You might want to check out Reuben Hersh’s What is Mathematics, Really? It is a whole book addressing exactly the point you raise.

First, math is definitely created. We invented some axioms and then started playing with them. This is basically the definition of creating something. We can (and have!) lay down other axioms and get other mathematical systems. We aren’t “finding” this stuff anywhere. We are inventing it.

Second, I don’t really understand your point about eternal truths. This doesn’t somehow disqualify math from being art. In fact, one could argue that this is something math has in common with great art. Aren’t the great classics exactly that because they reveal some human experience which is timeless? The existence of such timeless truths is what makes them great. We don’t disqualify them as art because there is something true in them…

No harm, no fowl. I had an inferiority complex long before you played the grad school card. I’ve read some Reuben Hersh, although not that title. Sounds interesting. I’ve studied enough to have seen some of the many effects of altering the axiomata. But wouldn’t the facts concerning which axiomatic systems yield which assortments of truths, falsehoods and undecidables be a static entity (susceptible more to discovery/exploration than invention) in its own right? It would seem so from my limited and limiting perspective, or is there some paradox or uncertainty or flat-out contradiction underlying the assumption of such a superset?

hilbertthm90

February 16, 2014 at 4:41 pm

I think in my attempt to be brief I haven’t done a very good job explaining what I meant. I only meant to use the axiom example as an example that a large portion of real math research involves creating definitions. In fact, most of math research involves these subjective types of choices that can’t possibly exist somewhere else because they are being created to best serve the mathematician’s purpose.

I understand your point. You basically want to say that because all theorems already “exist” as true/false/undecidable once you fix your system that these are being “found.” I just totally disagree. My analogy would be that this is like saying Rodin’s Thinker is found art because it “existed” in the stone it was carved from.

Just as the artistic value of a statue isn’t the stone but how the artist utilizes their tools to craft something from it, the artistic value of a theorem is not the truth/false/undecidedness but how the mathematician uses their tools to craft all the ideas around it. Just as an artist is “merely” rearranging existing things in a way that can reveal deep truths that you didn’t see before, a mathematician can reveal deep truths through their creative rearrangement of ideas.

It is kind of hard to succinctly and clearly say what I’m trying to say. I guess one could maybe make an argument that the truth/falseness of a proposition is “found,” but that seems to be somewhat tangential to what mathematicians actually do. If this truth/falseness is what we cared about, then there wouldn’t be so many proofs of Quadratic Reciprocity. The creative part is how you get to that truth. Since some of those proofs are beautiful and some are ugly, it can’t be merely the proposition itself that matters.

This seems to resonate well with the idea that it isn’t the truth/falseness of a theme in a novel which is the important artistic part, but the novel as a whole that shows the reader this truth/falseness. This is the same as when a mathematician writes a paper. There is context, language usage, figuring out how many vague ideas to explain and how many details to give, which definitions to give/use, and on and on which will determine its value.

Oops. I seem to be writing a whole blog post of my own now, so I’ll end it there.

While I agree with your general view of math research (yes, I am an actual researcher), I don’t think mathematics consists purely of creative formation. I think certain fundamental truths in mathematics are indeed there to be found.

The test that I use is: would an alien civilization completely removed from ours be likely to formulate an identical concept independently (up to notation changes)? For many of our mathematical achievements, the answer to this question would surely be “yes”, and if so, the situation is far closer to that of finding a common truth rather than crafting a unique (unreplicable) work of art.

So I think certain parts of math, like the values of Pi and e, the Pythagorean theorem, the statement of quadratic reciprocity, the monster group, and so on, are universal truths, and not so much created as found. Other parts of math, like the most elegant proof of a certain result, definitely qualify as pure creative art. Let us also remember that math is special precisely because it is probably the only academic discipline that spans both these spheres.

hilbertthm90

February 17, 2014 at 12:52 am

Yes. I didn’t mean to imply that what I wrote was the whole story. I just focused on that aspect to point out the sense in which math is creative.
On the other hand your thought experiment is much more subtle than most people initially think. Lakoff and Nunez have a book Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being that essentially argues that our version of math is intimately tied to our experience as embodied humans.
It is entirely plausible that if aliens experience the world in a different way that their math would be completely different and they wouldn’t necessarily have those things you list. It is just impossible for us to conceive how this could be as we are so limited to our own embodied minds.
I’m not saying I totally buy this, but the question of the universal nature of (our version of) math is not as open and shut a question as it looks like at first glance.

Swainson Holness

February 20, 2014 at 12:51 pm

I hate Jumping in like this but the above argument is so interesting. I have a lifetime of experiencing composing and playing music on the piano. There are only 88 keys with a finite set of combinations. Those combinations can be greatly varied with the timing that’s included. As I’ve learned these combinations and timing I’ve found that music is less “art” than I’d thought. This is because as more of these combinations are learn’t, patterns and rules reveal themselves to relate these combinations together in a “space” or theory. .

Looking at math from my high school education it was the opposite. It was taught that these rules were the “space” in which math was developed. I now have a hard time believing that because math is like music. The difference being that music has the piano which has 88 keys to be creative on and math the embodied mind. Taking that into account the “art” we find in math is the ability for the human mind to seek new assumptions. Its like extending the 88 keys on the piano, only the piano is the embodied mind.

For instance suppose that the axioms in any mathematical system can be added upon or modified in a way such that the original system is an equivalent subset of the modified system. Thus the assumptions that were modified or added extends the space in which the theorems of the original system relate with each other. This, for me, is the way in which art exists in mathematics. To discover irrational numbers for instance one has to make new or modify assumptions about how natural and rational numbers behave together. The weird thing about assumptions though is that there needs to be an embodied mind to make them.

In essence I agree with both points made in these arguments. Within the confines of the rules established routine mathematics is not artistic as much as a computer executing the midi mapping of a composition is not artistic. However when there is research to be done, at some point, there must be an idea, an assumption that comes not from the space of defined rules but from the space where ideas reside and that space is the human mind. Its the difference between a computer executing a midi piece and a person composing said piece. Obviously a computer can compose too but it cannot compose more than the conditions programmed into it. It is only the human mind at present that can do this and for me is a good place to define “art” in mathematics.

What Is Mathematics? -Richard Courant is another great book addressing some of these points.

Not sure how to persuade the larger public. The reason I favor more university-based research in general is because more published research compared to the amount of classified and/or proprietary research means a more transparent world, whereas letting the pendulum swing too far in the other direction (especially combined with failure to bring mathematical and scientific education to the masses) leads to a cargo cult, in which most people don’t understand the technologies on which their lives depend. The importance of stocking up humanity’s common body of knowledge goes triple for mathematics, because mathematics is a decoder ring to many other types of knowledge. If the NSA hogs up all the mathematical talent (or even if a large % of the academic mathematicians have been “clearanced” early in their careers) then maybe (or maybe not) the concept of humanity having a common body of mathematical knowledge is a polite fiction.

There is a tragedy of the commons at work because if research is funded primarily at the nation-state level (NSF, etc.) then each nation state wants technological supremacy, whereas all nations benefit from that which is published. In the case of privately-funded proprietary research, there is a tragedy of the anticommons: In the commercial knowledge sphere standing on the shoulders of giants requires your lawyers to contact the giants’ lawyers for patent clearance or whatever. So there must be a large international fund for basic research, one that hopefully places high priority on pure mathematics.

Speaking as one of the young people you say you’ve been discouraging (you haven’t talked to me specifically, but I’m the type of person you’re talking about; in particular, a recent HCSSiMer), I’m not fully convinced that I should give up on trying to be a professor or similar. In particular, I think the struggle to find a position doing research, for those who truly love mathematics, could be considered what you called “meaningful suffering” in an earlier post. Do you not see it this way, is my conception of “meaningful suffering” different from yours? (I wasn’t entirely sure, even after rereading the original post.) Or have you been saying things like, “Instead of starving while trying to be a math professor, you could do such-and-such and find a job where you have some amount of free time to do math research on your own, so you would actually get to do more math in the long run.”, or in other words “The way to optimize mathdoings is no longer professorship.”?

History seems to be full of long stretches of time in which almost all professional scholars are nuns or monks or something like that (not that that, in itself, constitutes suffering, of course). Might it be that one problem with adjunct hell is not the poverty, but the calculated and leveraged uncertainty as to whether one is scholarly material?

I think ‘because it’s deep’ is a more pertinent case for intrinsic value than ‘because it’s beautiful,’ even if the two concepts end up having the same extension. People easily accept fundamental physics research as important because it’s clear to them that fundamental physics is deep — that whatever it is that fundamental physics people do, it amounts to answering primordial questions everybody can acknowledge as important, even if not everybody can can follow the answers or even understand the fully precisified form of the questions. Fundamental physics research has the same kind of consensus value that e.g. going to the moon had: people find it intrinsically valuable for *someone*, not necessarily themselves, to be doing it, even if their own lives are not directly enriched by it. So I think the importance of math research can follow pretty easily from effectively making the case that mathematicians ask deep, rather than just complex, questions.

I think that in order to be convincing in your argument, you need specific relevant examples to illuminate the underlying reason. For example, the recent Science paper (http://news.sciencemag.org/technology/2014/02/termite-inspired-robots-build-bricks)
introduces a proof of concepts that robots can work together to build a structure without being given structural designs. Rather each can sense their environment and respond to a set of rules to complete the construction. This creates a novel mathematical problem of determining a set of rules for a given structure and collection of robots. One (sorry, I’m a mathematician. I do use that word more that I should.) could even think of proving theorems about metrics that determine whether a set of rules is complete or whether it is minimal. I think that this example is quite germaine for both points 1 and 3.

I also believe that we as mathematicians could do ourselves some favors by picking up problems like these and solving them (even if our primary area of research is stacks, or arithmetic geometry, or Teichmuller theory, or whatever). We ourselves are much too pigeonholed in our intellectual pursuits.

I do not understand your easy dismissal of your first reason. I think that there is a flaw there. There does not need to be a direct connection between the reason why I do mathematics and the reason why I get paid to do mathematics. So whilst I might say that I do mathematics because of the inherent beauty of the subject, I’m happy to be paid because _some_ mathematics proves useful. It may not be mine, but the people paying are prepared to pay me on the off-chance that it is and I’m prepared to take their money on that basis.

This seems an entirely valid justification of funding pure mathematics research. That Hardy didn’t see any use for his mathematics does not negate the fact that it is now used and has justified every penny he got in his salary.

It also justifies the _public_ funding of mathematics. The benefits are somewhat diffuse and it is hard to pinpoint exactly which bit of research will lead to the next Big Thing that makes it all worth it. So if private money were used, we may well miss out on things. Just as if private money were used to pay for road upkeep then there would be some places that wouldn’t get roads.

Your second reason may be closer to the reason why we do mathematics, but it doesn’t justify the public funding thereof. If that is the reason, let wealthy philanthropists fund it as it doesn’t bring any benefit to society. It doesn’t even have the saving grace of art in that no-one has ever said “I don’t know much about mathematics, but I know what I like.”. So as a justification for _public_ funding, this is the weakest of all.

Your third reason is a little undefined for me. I’m not clear as to whether you are talking about the mathematicians themselves or the others. I see no benefit to _society_ that mathematicians are trained to know when they are wrong (moreover, in my experience we are trained to know when we are wrong in a very specific field and are very bad at extending that to wider realms; indeed, one might even say that we are more pigheadded outside our own expertise and more reluctant to give up our daft ideas). We’re just too few and too ineffectual when it comes to influencing those who can actually make a change.

On the other hand, may be you are referring to the students that we teach. There it does make sense to talk about training people to think mathematically. True, we don’t do a great job of it, but at least we try! And it is good practice that those who teach a subject know a little bit more about it than they are trying to teach. So to get the best mathematical teaching at the university level, we need those involved in research to be teaching it (again, I’m actually not sure that this is true but that’s an argument for another blog post). Thus, we justify paying people to do mathematics research because otherwise they won’t teach our students. Once again, the reason why mathematicians teach need not be the reason why they are paid to teach, but that doesn’t negate the use of this as a justification.

Thus we argue that it is worth paying mathematicians to do research, even in to the purest of pure mathematics, on two bases:

1. It has happened in the past that even the purest mathematics is applicable to real problems, and provides real opportunities for making the world a better place. Thus the value of mathematics research _as a whole_ more than justifies the cost.

2. Mathematically educated people are better able to make sense of the world around them and so to make more sensible decisions on how to live in it. We want the best mathematicians to be teaching them, and they will only do so if we also allow them time to do their research.

Neither of these is likely to be the reason why a mathematician does their research, but what of it? So long as our interests are aligned, why does it matter?

jfepperson

February 21, 2014 at 9:59 am

I got in to math to solve “real problems.” (I’m a numerical analyst.) One thing I’ve learned in a 30+ year career, is that you cannot predict what the next Great Idea will be, nor how it might be applied. Number theory, traditionally the purest of the pure math fields, is now the underlying basis for making Internet transactions (sort of) secure. The scientists and engineers who “actually build things” need us to provide them with the tools they use. That is why math deserves more funding. Besides, we’re cheap: Some salary, a computer, some pads to write on—that’s all we need. No expensive lab!

Hello, I’m an undergraduate at Illinois Tech, who has gone back to school to get a B.S. in Applied Mathematics. So by no means do I pretend that my perspective on this matter is at all fully developed. That being said, I think the fundamental argument for perpetuating funding for “pure mathematics” research (AMAT is a different beast) is rather straight forward. That is, “pure mathematics” is the bed-rock from which, scientific discovery has come. More specifically, the development of “pure mathematics” is the means by which, abstract thoughts are formalized, made coherent, and turned into useful tools for the rest of science to use. To better understand what I mean by this, I ask that you look at the history and the process from which, scientific has come. If one does so, one will most likely find a trend or loose narrative. That narrative may be paraphrased as follows: philosophic ideas are imagined -> a loose reasoning and logic are developed around these ideas -> a mathematics is developed to more clearly define the reasoning and logic of the ideas -> a scientific field finds these ideas useful and they are applied to new or existing scientific models.
In this narrative, mathematics is the turning point of the story where the abstract becomes useful. Now I will admit that this narrative is not a contemporary one, but rather the root story of science. With the invention of the computer, the turn around for making abstract ideas become applied ideas has increased significantly. Today with applied mathematics, computer science, physics, biology, etc… the process of days of old has been turned into a system where the mathematics is simultaneously developed along side their respective scientific fields. Never the less, the freedom to develop purely mathematical constructs continues to be a means for developing the fundamental tools to later be applied to a whole host of scientific fields. To only develop mathematics as the need for them arises in a specific scientific field is very short sighted and could potentially limit scientific discovery. Or even worse, abort the creation of new scientific fields before they are born. Because, after all, pure mathematics has proved itself to be the fertile ground from which most scientific fields have come.
With one caveat, if mathematics hopes keep up with the increasing speed that knowledge is developed, it must learn to adapt and more fully adopt the use of modern tools, such as the computer. Kids today are taught to use computers from a very young age. As a result they are able to learn and discover things much quicker than previous generations. So the rigid mathematical paradigm of old where one strictly uses only their brain to understand mathematical concepts is quickly becoming an archaic one. I would say that if mathematicians desire themselves to be viewed as being useful, they must adopt tools and practices which allow them to keep up with the speeds of development and transfer of information that society has become accustom to.

Cathy, I’m a mathbio postdoc with an applied math background and some background in biology (htttp://www.pauljhurtado.com/) and if I understand you correctly, what you’re calling “mathematics research” is really “pure mathematics research”, i.e., it’s distinct from applied math research, that uses existing (or new) mathematics to tackle scientific or techological problems.
If that’s the case, you should definitely be explicit about pure mathematics versus mathematics sensu lato — because there are very different justifications for supporting research in pure vs. applied math! Namely, the “applied research” sell is easy — the ends justify the means. So that leaves arguments for investing in pure math research…
A nice parallel to consider here is “science research” which is often similarly split into “basic” and “applied” areas of research. Drawing that parallel, I’d encourage digging up arguments for funding basic science over applied science — they tend to be very applicable to thinking about funding pure vs applied mathematics research.
So yeah — I’d add the following argument to your list: funding pure mathematics research is an investment in maintaining and advancing the intellectual infrastructure that forms the foundation of applied mathematics research, which itself is having a huge impact on science and technology research that directly impacts nearly every facet of human existence. Simply put, pure problems have a track record for being (or eventually leading to) more directly useful mathematical machinery.

marj

March 11, 2014 at 2:35 am

” Continuing math research is important because it trains people to think abstractly and to have a skeptical mindset.” This is critical to develop one’s sense of rational skepticism. Marl of Mymathdone.com